Add $-4x-1$ and $ \dfrac{x^4}{10} $ to get $ \dfrac{ \color{purple}{ x^4-40x-10 } }{ 10 }$.

Step 1: Write $ -4x-1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 10 }$.

$$ \begin{aligned} -4x-1+ \frac{x^4}{10} & \xlongequal{\text{Step 1}} \frac{-4x-1}{\color{red}{1}} + \frac{x^4}{10} = \frac{ \left( -4x-1 \right) \cdot \color{blue}{ 10 }}{ 1 \cdot \color{blue}{ 10 }} + \frac{ x^4 }{ 10 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -40x-10 } }{ 10 } + \frac{ \color{purple}{ x^4 } }{ 10 }=\frac{ \color{purple}{ x^4-40x-10 } }{ 10 } \end{aligned} $$

Subtract $2x^2$ from $ \dfrac{x^4-40x-10}{10} $ to get $ \dfrac{ \color{purple}{ x^4-20x^2-40x-10 } }{ 10 }$.

Step 1: Write $ 2x^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{10}$.

$$ \begin{aligned} \frac{x^4-40x-10}{10} -2x^2 & \xlongequal{\text{Step 1}} \frac{x^4-40x-10}{10} - \frac{2x^2}{\color{red}{1}} = \frac{ x^4-40x-10 }{ 10 } - \frac{ 2x^2 \cdot \color{blue}{ 10 }}{ 1 \cdot \color{blue}{ 10 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4-40x-10 } }{ 10 } - \frac{ \color{purple}{ 20x^2 } }{ 10 }=\frac{ \color{purple}{ x^4-20x^2-40x-10 } }{ 10 } \end{aligned} $$